# For what values of $a$ will this function be continuous for all $x$?

Let:

$$g(x)=\frac{1}{1+e^{1/(x-1)}}$$

for $x\ne 1$ and $g(x)=a$ for $x=1$.

For what values of $a$ will $g(x)$ be continuous for every $x$?

• See here for a reference to make the math in your question much more readible. Dec 28 '12 at 12:21
• Will apply this from now on. Thanks.
– pie
Dec 28 '12 at 12:24

Hint Look at $\lim\limits_{x\rightarrow1^+}g(x)$ and $\lim\limits_{x\rightarrow1^-}g(x)$ to determine what $a$ needs to be.

• But since limg(x) from the right is different from the left, can the function ever be continues for all x? Even if a=1, then function still isn't continues.
– pie
Dec 28 '12 at 12:28
• @pie If the two one-sided limits are not equal, what can you say about the two-sided limit, and hence the continuity of the function at that point? Dec 28 '12 at 12:32
• The two sided limit doesn't exist - but then the questions asks "For what values of $a$ will g(x) be continues for all x?".
– pie
Dec 28 '12 at 12:34
• @pie Looking at the graph here it is immediately obvious that no value of $a$ will make the function continuous at $x=1$. The limit analysis that you did confirms this conclusion. Dec 28 '12 at 12:38
• Thank you, what a terrible wording that question has.
– pie
Dec 28 '12 at 12:48